Diffusions, Markov Processes, and Martingales Volume 2: ITO 2nd Edition CALCULUS L. C. G. ROGERS School of Mathematical Sciences, University of Bath and DAVID WILLIAMS Department of Mathematics, University of Wales, Swansea CAMBRIDGE UNIVERSITY PRESS
Contents Some Frequently Used Notation xiv CHAPTER IV. INTRODUCTION TO ITO CALCULUS TERMINOLOGY AND CONVENTIONS R-processes and L-processes Usual conditions, etc. Important convention about time 0 1. SOME MOTIVATING REMARKS 1. Ito integrals 2 2. Integration by parts 4 3. Ito's formula for Brownian motion 8 4. A rough plan of the chapter 9 2. SOME FUNDAMENTAL IDEAS: PREVISIBLE PROCESSES, LOCALIZATION, etc. Previsible processes 5. Basic integrands Z(S, 7] 10 6. Previsible processes on (0, oo), ^*, bj*, W 11 Finite-variation and integrable-variation processes 7. FV 0 and IV 0 processes 14 8. Preservation of the martingale property 14 Localization 9. H(0, r], X T 15 10. Localization of integrands, lb^* 16 11. Localization of integrators, J? 0,\ x, FVj? 04oc etc 17 12. Nil desperandum! 18 13. Extending stochastic integrals by localization 20 14. Local martingales, ~# loc, and the Fatou lemma 21 Semimartingales as integrators 15. Semimartingales,^ 23 16. Integrators 24 Likelihood ratios 17. Martingale property under change of measure 25
IX 3. THE ELEMENTARY THEORY OF FINITE-VARIATION PROCESSES 18. Ito's formula for FV functions 27 19. The Doleans exponential r (x.) 29 Applications to Markov chains with finite state-space 20. Martingale problems 30 21. Probabilistic interpretation of Q 33 22. Likelihood ratios and some key distributions 37 4. STOCHASTIC INTEGRALS: THE L 2 THEORY 23. Orientation 42 24. Stable spaces of Jt\, cjt\, &M\ 42 25. Elementary stochastic integrals relative to M in M\ 45 26. The processes [Af ] and [M,,/V] 46 27. Constructing stochastic integrals in L 1 47 28. The Kunita-Watanabe inequalities 50 5. STOCHASTIC INTEGRALS WITH RESPECT TO CONTINUOUS SEMIMARTINGALES 29. Orientation 52 30. Quadratic variation for continuous local martingales.... 52 31. Canonical decomposition of a continuous semimartingale... 57 32. Ito's formula for continuous semimartingales 58 6. APPLICATIONS OF ITO'S FORMULA 33. Levy's theorem 63 34. Continuous local martingales as time-changes of Brownian motion 64 35. Bessel processes; skew products; etc 69 36. Brownian martingale representation 73 37. Exponential semimartingales; estimates 75 38. Cameron-Martin-Girsanov change of measure 79 39. First applications: Doob /i-transforms; hitting of spheres; etc.. 83 40. Further applications: bridges; excursions; etc 86 41. Explicit Brownian martingale representation 89 42. Burkholder-Davis-Gundy inequalities 93 43. Semimartingale local time; Tanaka's formula 95 44. Study of joint continuity 99 45. Local time as an occupation density; generalized Ito-Tanaka formula 102 46. The Stratonovich calculus 106 47. Riemann-sum approximation to Ito and Stratonovich integrals; simulation 108
X CHAPTER V. STOCHASTIC DIFFERENTIAL EQUATIONS AND DIFFUSIONS 1. INTRODUCTION 1. What is a diffusion in 1^"? 110 2. FD diffusions recalled 112 3. SDEs as a means of constructing diffusions 113 4. Example: Brownian motion on a surface 114 5. Examples: modelling noise in physical systems 114 6. Example: Skorokhod's equation 117 7. Examples: control problems 119 2. PATHWISE UNIQUENESS, STRONG SDEs, AND FLOWS 8. Our general SDE; previsible path functionals; diffusion SDEs. 122 9. Pathwise uniqueness; exact SDEs 124 10. Relationship between exact SDEs and strong solutions.... 125 11. The Ito existence and uniqueness result 128 12. Locally Lipschitz SDEs; Lipschitz properties of a 1 ' 2.... 132 13. Flows; the diffeomorf -,n theorem; time-reversed flows... 136 14. CarverhilFs noisy North-South flow on a circle 141 15. The martingale optimality principle in control 144 3. WEAK SOLUTIONS, UNIQUENESS IN LAW 16. Weak solutions of SDEs; Tanaka's SDE 149 17. 'Exact equals weak plus pathwise unique' 151 18. Tsirel'son's example 155 4. MARTINGALE PROBLEMS, MARKOV PROPERTY 19. Definition; orientation 158 20. Equivalence of the martingale-problem and'weak'formulations 160 21. Martingale problems and the strong Markov property.... 162 22. Appraisal and consolidation: where we have reached.... 163 23. Existence of solutions to the martingale problem 166 24. The Stroock-Varadhan uniqueness theorem 170 25. Martingale representation 173 Transformation of SDEs 26. Change of time scale; Girsanov's SDE 175 27. Change of measure 177 28. Change of state-space; scale; Zvonkin's observation; the Doss- Sussmann method 178 29. Krylov's example 181 5. OVERTURE TO STOCHASTIC DIFFERENTIAL GEOMETRY 30. Introduction; some key ideas; Stratonovich-to-Ito conversion. 182 31. Brownian motion on a submanifold of B N 186
XI 32. Parallel displacement; Riemannian connections 193 33. Extrinsic theory of BM h0r (O( )); rolling without slipping; martingales on manifolds; etc 198 34. Intrinsic theory; normal coordinates; structural equations; diffusions on manifolds ;etc.(!) 203 35. Brownian motion on Lie groups 224 36. Dynkin's Brownian motion of ellipses; hyperbolic space interpretation; etc 239 37. Khasminskii's method for studying stability; random vibrations. 246 38. Hormander's theorem; Malliavin calculus; stochastic pullback; curvature 250 6. ONE-DIMENSIONAL SDEs 39. A local-time criterion for pathwise uniqueness 263 40. The Yamada-Watanabe pathwise uniqueness theorem.... 265 41. The Nakao path wise-uniqueness theorem 266 42. Solution of a variance control problem 267 43. A comparison theorem 269 7. ONE-DIMENSIONAL DIFFUSIONS 44. Orientation 270 45. Regular diffusions 271 46. The scale function, s 273 47. The speed measure, m; time substitution 276 48. Example: the Bessel SDE.' 284 49. Diffusion local time :.... 289 50. Analytical aspects 291 51. Classification of boundary points 295 52. Khasminskii's test for explosion 297 53. An ergodic theorem for 1-dimensional diffusions 300 54. Coupling of 1-dimensional diffusions 301 CHAPTER VI. THE GENERAL THEORY 1. ORIENTATION 1. Preparatory remarks 304 2. Levy processes 308 2. DEBUT AND SECTION THEOREMS 3. Progressive processes 313 4. Optional processes, (?; optional times 315 5. The 'optional' section theorem 317 6. Warning (not to be skipped) 318
Xll 3. OPTIONAL PROJECTIONS AND FILTERING 7. Optional projection X of X 319 8. The innovations approach to filtering 322 9. The Kalman-Bucy filter 327 10. The Bayesian approach to filtering; a change-detection filter.. 329 11. Robust filtering 331 4. CHARACTERIZING PREVISIBLE TIMES 12. Previsible stopping times; PFA theorem 332 13. Totally inaccessible and accessible stopping times 334 14. Some examples 336 15. Meyer's previsibility theorem for Markov processes 338 16. Proof of the PFA theorem 340 17. The ff-algebras f(p-),f(p), J*"(P + ) 343 18. Quasi-left-continuous nitrations 346 5. DUAL PREVISIBLE PROJECTIONS 19. The previsible section theorem; the previsible projection P X of A" 347 20. Doleans' characterization of FV processes 349 21. Dual previsible projections, compensators 350 22. Cumulative risk 352 23. Some Brownian motion examples 354 24. Decomposition of a continuous semimartingale 358 25. Proof of the basic (/i, -4) correspondence 359 26. Proof of the Doleans 'optional' characterization result.... 360 27. Proof of the Doleans 'previsible' characterization result... 361 28. Levy systems for Markov processes 364 6. THE MEYER DECOMPOSITION THEOREM 29. Introduction 367 30. The Doleans proof of the Meyer decomposition 369 31. Regular class (D) submartingales; approximation to compensators 372 32. The local form of the decomposition theorem 374 33. An L 2 bounded local martingale which is not a martingale.. 375 34. The <M> process 376 35. Last exits and equilibrium charge 377 7. STOCHASTIC INTEGRATION: THE GENERAL CASE 36. The quadratic variation process [M] 382 37. Stochastic integrals with respect to local martingales.... 388 38. Stochastic integrals with respect to semimartingales 391 39. Ito's formula for semimartingales 394 40. Special semimartingales 394 41. Quasimartingales 396
Xlll 8. ITO EXCURSION THEORY 42. Introduction 398 43. Excursion theory for a finite Markov chain 400 44. Taking stock 405 45. Local time L at a regular extremal point a 406 46. Some technical points: hypotheses droites, etc 410 47. The Poisson point process of excursions 413 48. Markovian character of n 416 49. Marking the excursions 418 50. Last-exit decomposition; calculation of the excursion law n.. 420 51. The Skorokhod embedding theorem 425 52. Diffusion properties of local time in the space variable; the Ray-Knight theorem 428 53. Arcsine law for Brownian motion 431 54. Resolvent density of a 1-dimensional diffusion 432 55. Path decomposition of Brownian motions and of excursions.. 433 56. An illustrative calculation 438 57. Feller Brownian motions 439 58. Example: censoring and reweighting of excursion laws.... 442 59. Excursion theory by stochastic calculus: McGill's lemma... 445 REFERENCES 449 INDEX 469